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Clancy, P. & Randolph, M. F. (1996). GeÂotechnique 46, No. 2, 313±328

Simple design tools for piled raft foundations P. C L A N C Y  a n d M . F. R A N D O L P H  La conception des fondations par radier fonde sur pieux est rendue dif®cile par les interactions complexes qui existent entre les pieux et le radier. Bien que les meÂthodes par eÂleÂments ®nis et par eÂleÂments limites aient eÂte utiliseÂes pour prendre en compte de manieÁre explicite ces interactions, l'emploi de ces meÂthodes est limite aux systeÁmes de fondations aÁ petits groupes de pieux. La meÂthode hybride deÂveloppeÂe permet d'analyser compleÁtement des systeÁmes de fondations plus importants (en utilisant la symeÂtrie d'ordre 4). En routine, la conception d'un radier fonde sur pieux multiples reÂsulte de l'extrapolation des reÂsultats obtenus pour un radier fonde sur un seul pieu. Ceci demande forcement quelques approximations. On s'est eÂgalement un peu inteÂresse aÁ la deÂtermination des tassements diffeÂrentiels qui jouent souvent un roÃle critique lors de la conception. L'article deÂcrit brieÁvement la meÂthode hybride {eÂleÂments ®nis-milieu eÂlastique} qui a eÂte deÂveloppeÂe pour les radiers fondeÂs sur pieux, ainsi qu'une meÂthode approcheÂe deÂjaÁ existante permettant de calculer, en tout point, la raideur de la fondation et la distribution de la charge. La meÂthode approcheÂe a pu eÃtre ameÂlioreÂe en comparant les reÂsultats obtenus aÁ ceux de la meÂthode hybride. Des eÂtudes parameÂtriques, reÂaliseÂes aÁ l'aide de la meÂthode hybride, ont permis le deÂveloppement de meÂthodes approcheÂes permettant d'estimer les tassements diffeÂrentiels. On preÂsente ensuite un exemple d'application des meÂthodes approcheÂes. Les deux cas-types de fondation par radier fonde sur pieux proposeÂs montrent comment utiliser l'une des meÂthodes approcheÂes et permettent de comparer les reÂsultats obtenus aÁ ceux issus d'une analyse compleÁte et aux mesures de chantier.

The design of piled raft foundations is hampered by the complex interaction effects which take place between the piles and the raft. Although the ®nite element method and the boundary element method have been used to account explicitly for the interactions, the application of these methods is limited to foundation systems with small pile groups. A hybrid method of analysis has been developed, by which the complete analysis of larger foundation systems (using quadrant symmetry) may be performed. More practical approaches for the routine design of piled rafts rely on the extrapolation of results produced for single pile± raft units, but necessarily incorporating approximations. Little attention has been paid to the determination of differential settlements, which are often critical to the design. The paper outlines the hybrid ®nite element±elastic continuum method of piled raft analysis, and describes an approximate method for calculating overall foundation stiffness and load distribution. The approximate method may be re®ned by comparison with results from the hybrid method. Parametric studies using the hybrid method have been used to develop approximate methods for estimating differential settlements. An example of the application of the approximate methods is presented. Two case studies of piled raft foundations are made, which show the application of one of the approximate methods and compare the results with a complete analysis and with ®eld measurements.

KEYWORDS: design; footings/foundations; numerical modelling and analysis; piles; rafts; settlement.

and raft±soil contact stresses. Such foundations occur frequently in practice for two main reasons: ®rst, even when a foundation has been designed as a pile group it is common practice to cast the cap of the pile group directly on the ground; second, the settlements of a raft foundation may be reduced by the installation of piles. The current lack of ef®cient, reliable design methods for piled rafts has resulted in a design practice that commonly

INTRODUCTION

Piled raft foundations transmit structural loads to the soil by way of both pile±soil contact stresses Manuscript received 25 May 1994; revised manuscript accepted 10 May 1995. Discussion on this paper closes 2 September 1996; for further details see p. ii.  University of Western Australia.

313

314

CLANCY AND RANDOLPH

ignores the bearing contribution of the raft. This results in a conservative estimate of the foundation performance, and therefore an overdesign of the foundation. Numerical analysis of piled raft foundations presents a major problem because of the computational resources required for foundations of practical proportions. Few methods of analysis can allow for the load-carrying potential of the raft explicitly, and most of these are either very approximate or of limited applicability. Davis & Poulos (1972) used the boundary element method in a parametric study of pairs of interacting pile± raft units. A superposition approach, similar to that used previously for pile groups (Poulos, 1968), was adopted for larger numbers of piles. Randolph (1983) presented a method that combined the individual `stiffnesses' of pile group and raft, based on the study of a single representative pile±raft unit. A more rigorous study was presented by Ottaviani (1975) using three-dimensional ®nite elements, but this was limited to a maximum of 15 piles. Later a hybrid ®nite element±elastic continuum method (Hain & Lee, 1978) was developed, enabling the consideration of a 6 3 6 piled raft. Grif®ths, Clancy & Randolph (1991) presented a hybrid ®nite element±elastic continuum±load transfer approach which was developed speci®cally to minimize the amount of computation. The latest development of this method takes advantage of quadrant symmetry to allow the complete analysis of piled rafts with more than 200 piles. Even with the development of hybrid methods the applicability of complete numerical analyses to real problems remains limited, due to the magnitude of computer resources required in the analysis of larger foundation systems. In order to allow the routine design of piles raft foundations, it is necessary to develop approximate methods that allow extrapolation of the rigorous analyses. The present work draws on the results of an extensive parametric study using the hybrid method of Grif®ths et al. (1991) as the basis for the development of simple design tools. The following sections describe each of the methods in turn, before comparing results directly. HYBRID APPROACH

The hybrid ®nite element±elastic continuumload transfer approach has been described in detail elsewhere (Grif®ths et al., 1991; Clancy & Randolph, 1993; Clancy, 1993). The major features of the method are presented in Fig. 1. Onedimensional rod ®nite elements (Smith & Grif®ths, 1988) are used to model the piles, while the pile± soil contact is represented at node points by

6 3 4

7 1 2 5

Fig. 1. Numerical representation of piled raft foundation: (1) one-dimensional pile element; (2) lumped soil response at each pile nodeÐload-transfer spring; (3) two-dimensional plate-bending ®nite element raft mesh; (4) lumped soil response at each raft nodeÐ Giroud solution; (5) pile±soil±pile interaction effects calculated between pairs of nodesÐMindlin's equation; (6) raft±soil±raft interaction; (7) pile±soil±raft interaction

potentially non-linear load transfer springs (Randolph, 1977; Chow, 1986). The raft is subdivided into two-dimensional `thin' plate-bending ®nite elements (Smith & Grif®ths, 1988), and the raft±soil contact is lumped into an equivalent soil `spring' (Giroud, 1968) at each node. Finally, interaction effects between all pairs of nodes are calculated using the elastic solution of Mindlin (1936) in a point-to-point fashion. The hybrid method provides a relatively rigorous and yet considerably more ef®cient method of analysis for piled rafts than has previously been available. Fewer equations need to be solved than for the ®nite element method, and the timeconsuming numerical integrations of the boundary element method are not necessary. The hybrid method allows for variable geometry, pile stiffness, soil stiffness and raft stiffness. Although only vertical applied loading and linear elastic soil conditions are considered here, it is relatively straightforward to allow for non-linear response at the pile±soil interface, or to extend the analysis to include horizontal or inclined loading. DEVELOPMENT OF APPROXIMATE METHODS

Pile group±raft interaction The piled raft analysis proposed by Randolph (1983), which employs a `¯exibility' matrix to combine the individual `stiffnesses' (i.e. load± displacement response) of the pile group and raft, is attractively simple in its application. The method allows the overall stiffness and load distribution of

315

DESIGN TOOLS FOR PILED RAFTS

a piled raft foundation to be calculated by estimating the interaction effects between its component parts. Randolph's original work made an analytical study of a single pile±raft unit to derive k p ‡ k r (1 ÿ 2árp ) 1 ÿ (k r =k p )árp 2

(1)

(1 ÿ árp )k r Pr ˆ Pr ‡ Pp k p ‡ k r (1 ÿ 2árp )

(2)

k pr ˆ

load-transfer approach in the very short pile range (Lp /dp , 5), of relevance to equivalent piers rm ˆ 2
5rLp (1 ÿ ís ) ‡ 2
5d p

where kpr is the overall stiffness (load/displacement) of a piled raft, kp and kr are the overall stiffness of a pile group and of a raft in isolation respectively, Pp and Pr are the load carried by a pile group and by a raft in the complete foundation respectively, and árp is a factor quantifying the in¯uence of a pile group on the raft. Superposition of the displacement ®elds induced by a single pile and by a circular raft provided a rational basis for the calculation of árp árp ˆ 1 ÿ

ln (n) ln (2rm =d p )

(4)

It was assumed that the value of árp for pile groups could be calculated by considering a representative single pile±raft unit, where the representative unit has a raft area equal to the mean raft area per pile of the complete foundation. Clancy & Randolph (1993) explored the above approach for square piled rafts with properties as given in Tables 1 and 2, containing up to 36 piles, and concluded that values of árp tended towards 0´8 for larger pile groups. Taking advantage of quadrant symmetry, 12 3 12 pile groups may now be analysed using the same computational resources that were previously required for 6 3 6 pile groups. This allows a more extensive investigation of the convergence of árp with increasing pile group size. Fig. 2 shows the results of such a study, demonstrating that the convergence limit should be revised slightly to give árp = 0´85 for large pile groups, leading to expressions for the overall stiffness and load distribution

(3)

1 ÿ 0
7(k r =k p ) kp 1 ÿ 0
723(k r =k p )

(5)

Pr 0
15(k r =k p ) ˆ Pr ‡ Pp 1 ÿ 0
7(k r =k p )

(6)

Pp Pr ˆ1ÿ Pr ‡ Pp Pr ‡ Pp

(7)

k pr ˆ

where n is the ratio of circular raft diameter to pile diameter dp, and rm is the radius of in¯uence of a pile rm ˆ 2
5rLp (1 ÿ ís ) Lp is the embedded length of a pile, ís is the Poisson's ratio of soil and r is a soil inhomogeneity factor. Subsequently, Randolph (1994) suggested a modi®cation to the value of rm to improve the

Pile group ef®ciency charts (such as those published by Fleming, Weltman, Randolph & Elson

Table 1. Parameters for piled raft foundations Soil Young's modulus Es Poisson's ratio ís

Pile Young's modulus Ep Length Lp Diameter dp Spacing sp

Raft Young's modulus Er Poisson's ratio ír Thickness tr Length Lr Breadth Br

Table 2. Dimensionless groups for piled raft foundations Dimensionless group

De®nition

Pile slenderness ratio Pile spacing ratio Pile±soil stiffness ratio Raft plan aspect ratio Raft±soil stiffness ratio

Krs

Lp /dp sp /dp Kps = Ep /Es Lr /Br 4Er Br tr 3 (1 ÿ ís 2 ) = 3ðEs Lr 4 (1 ÿ ír 2 )

Practical range 10±100 2´5±8 100±10 000 1±10 0´001±10

316

CLANCY AND RANDOLPH 0.900

0.800

αrp

0.700

0.600

0.500

0.400 2

3

4

5

6

7

8

9

Pile spacing sp/dp

1 x 1 pile group

2 x 2 pile group

3 x 3 pile group

4 x 4 pile group

5 x 5 pile group

6 x 6 pile group

7 x 7 pile group

8 x 8 pile group

9 x 9 pile group

10 x 10 pile group

11 x 11 pile group

12 x 12 pile group

Fig. 2. árp values: Lp /dp = 25; Kps = 1000; Krs = 10

(1992) and Butter®eld & Douglas (1981)) may be used to estimate kp , the overall stiffness of the pile group. Closed-form analytical solutions for either circular or rectangular rafts are available (Poulos & Davis, 1974), from which the overall raft stiffness kr may be derived. This approximate method enables the straightforward calculation of load distribution and average settlement for a piled raft foundation, but does not address the key problem of differential settlements. Differential settlements The differential settlement of rafts and pile groups was considered by Randolph & Clancy (1993). For convenience in plotting the settlement pro®les of rafts, a co-ordinate system was introduced as shown in Fig. 3. Generally there are two points of interest for differential settlements relative to the centre of a rectangular raft: the corner and the mid-side. Thus, differential settlements may be referred to as 0´0±1´0 (corner±centre) and 0´5±1´0 (mid-side±centre). Differential settlements are normalized here by the mean raft settlement, which is relatively independent of raft stiffness. For raft foundations, the normalized differential settlement of circular and square rafts is a function of the relative raft stiffness Krs . For pile groups, it was found that the

0.5

0.0

1.0

cL

cL

Fig. 3. Normalized co-ordinates for rectangular rafts

clearest pattern was to plot the normalized differential settlement against a factor R, given by  r np sp (8) Rˆ Lp where np is the total number of piles in a group. The factor R combines a measure of the overall

317

DESIGN TOOLS FOR PILED RAFTS

aspect ratio of the group, de®ned as the total breadth divided by the pile length (i.e. Ï(np )sp /Lp ), and the degree of interaction of the piles, broadly quanti®ed by the ratio Ï(Lp /sp ). It was demonstrated that for large values of R (. 3±4), where the geometry of the pile group resembles that of a `shallow' foundation, the normalized differential settlement approaches that of a fully ¯exible raft foundation. For large piled raft foundations, it is likely that the geometry will lead to a high value of R. As demonstrated by Clancy & Randolph (1993), the absolute differential settlements of a piled raft may then be obtained from an analysis of the raft alone, but factored by the ratio kr /kpr to account for the overall reduction in settlements due to the presence of the piles in the complete foundation. To determine the settlement reduction factor, a knowledge of the gross piled raft behaviour (i.e. the value of kpr ) is assumed. The method, referred to here as the `combined pile group and raft' approach, therefore represents a natural extension to the approximate pile group±raft interaction approach described in the previous section. Raft and equivalent piers Poulos & Davis (1980) proposed the equivalent pier method for predicting the load±displacement response of a pile group. It was suggested that the overall behaviour of a pile group could be modelled by considering a single `equivalent pier' of effectively composite pile±soil material. The analysis of the resulting single pier is then straightforward. To derive appropriate properties for the pier, it was assumed that the length of the pier is the same as that of the piles in the group. This assumption results in an expression for the radius of the equivalent pier (Poulos, 1993) p rpeq ˆ Ag =ð (9) where Ag is the plan area of a pile group. The equivalent pier modulus may now be calculated according to   Ap Ap Epeq ˆ Ep ‡ Es 1 ÿ (10) Ag Ag where Ap is the sum of pile cross-sectional areas.

The equivalent pier approach facilitates the analysis of large pile groups, by replacing the full number of piles by a smaller number of equivalent piers. The modelling of several piers beneath the raft (rather than replacing the complete group by a single pier) allows an estimate of differential settlement. COMPARISON OF RESULTS

To investigate the effectiveness of the proposed approximate methods, a square 10 3 10 piled raft system was analysed. The problem is small enough to be analysed as a complete pile±raft system by the present hybrid method, and large enough for a meaningful application of the approximate methods. A basic set of material and geometrical properties was selected to represent typical ®eld values (Table 3). The appropriate parameters of this basic property set were modi®ed in turn to investigate the in¯uence of Kps , Krs , Lp /dp and sp /dp . The raft meshes used in the analyses are illustrated in Fig. 4; for the factored raft method, a similar level of mesh re®nement was used to that of the complete analysis. In each case, ten rod ®nite elements were used to model the piles or piers. A 5 3 5 pier group was used to replace the 10 3 10 pile group, reducing the problem by a factor of four. Table 4 gives the values of rpeq and Epeq calculated for the equivalent piers and also values of normalized stiffness (kp /np Es dp ) for each pile sub-group and corresponding equivalent pier. The equivalent pier approximation shows a tendency to overestimate stiffness by 5%±10%. Where equivalent piers are used to represent larger numbers of individual piles, the overestimation of stiffness may increase to up to 20% (Randolph, 1994). Gross foundation behaviour Figure 5 summarizes the predictions of overall piled raft behaviour in terms of the normalized mean settlement of the raft and the load distribution between the raft and the pile group. Excellent comparisons were obtained for the normalized mean settlement, while the load sharing predictions were reasonable. The combined pile group and raft

Table 3. Basic property set for comparison of piled raft behaviour Soil Es = 35 MPa ís = 0´4

Pile Ep Lp dp sp

= 35 000 MPa = 20 m = 0´8 m =4m

Raft Er ír tr Lr Br

= 35 000 MPa = 0´3 = 2´537 m = 40 m = 40 m

Dimensionless groups Lp /dp sp /dp Kps Lr /Br Krs

= 25 =5 = 1000 =1 = 0´1

318

CLANCY AND RANDOLPH cL

cL

sp = 4 m cL

Lr /2 = Br /2 = 20 m

cL

10 x 10 piled raft, d p = 0.8 m

Raft with 5 x 5 equivalent piers, d peq = 4.52 m

Fig. 4. Raft meshes for evaluation of piled raft approximations: basic geometry set, sp /dp = 5, quadrant symmetry

Table 4. Properties and normalized stiffnesses of equivalent piers (knorm = kp 3 1/np Es dp ) Pile group properties

rpeq : m

Epeq : MPa

2 3 2 pile group stiffness

Equivalent pier stiffness

Difference: %

Basic property set (Table 3) Kps = 100 Kps = 10 000 Kps = 100 000 Lp /dp = 10 Lp /dp = 100 sp /dp = 2´5 sp /dp = 7´5

2´26

4´427 3 103

5´62

5´98

+6´6%

2´26 2´26 2´26 2´26 2´26 1´13 3´39

4´703 4´400 4´397 4´427 4´427 17´61 1´987

103 103 103 103 103 103 103

3´78 5´98 6´03 3´68 9´60 4´74 6´28

3´75 6´45 6´50 3´98 9´47 4´63 7´05

20´7% +7´8% +7´8% +8´3% 21´2% 22´4% +12´1%

3 3 3 3 3 3 3

approach tends to underestimate the proportion of load transferred to the pile group, and conversely for the equivalent piered raft approach. The differences are typically less than 10% of the total applied load. Overestimation of the proportion of load carried by the piles using the equivalent piered raft approach is due partly to the tendency for the equivalent pier to overestimate pile subgroup stiffness, and partly to the additional raft area encompassed by the equivalent piers. The apparent underestimation of the loads taken by the piles, using the combined pile group and raft approach, is partly due to the level of mesh re®nement used for the raft in the full analysis. The numerical method assumes that no load is transferred to the soil by the raft within one quarter of the area of each raft element adjacent to a pile. Thus, the use of only two raft elements over ®ve pile diameters

overestimates the load transferred to the soil by the piles. Use of a ®ner mesh, as shown in Fig. 6, has little in¯uence on the mean settlement, but the load distribution is altered so that a greater proportion of the total load is transferred to the soil via raft contact stress (see Fig. 5). Differential settlement Normalized settlement pro®les for the 10 3 10 piled raft are shown in Fig. 7 for a range of raft± soil stiffness ratios. The results from both of the approximate methods are plotted along with those from a complete piled raft analysis, and all three methods show good agreement, with errors in the estimated differential settlements for the more ¯exible rafts being less than 5% of the mean settlement. Additional studies have been undertaken to

319

1

1

0.8 0.6

0.8 0. 6

Pp / P , P r / P

w pr*

DESIGN TOOLS FOR PILED RAFTS

0.4 0.2

0. 4 0.2 0

0 2

3

4

2

5

3

0.4 0.2 −2

1

0.4 0.2

−1

0

−2

1

log 10 Krs

−1 log 10 Krs

1

1

0.8 0.6

0.8 0. 6

Pp / P , P r / P

w pr*

0

0.8 0.6

0

0

0.4 0.2 0

0. 4 0.2 0

0

50

100

0

L p /d p

1

1

0.8 0.6

0.8 0. 6

0.4 0.2 2.5

5

w p*r = w pr

100

7.5

0. 4 0.2 0

s p /d p

LrEs x [P(1 − υs2)]

50

L p /d p

Pp / P , P r / P

w pr*

5

1

1 0.8 0.6

0

4 log 10 Kps

Pp / P , P r / P

w pr*

log 10 Kps

2.5

5

s p /d p

Piled raft analysis: w pr*, Pp /P Piled raft analysis: P r /P Combined pile group and raft Raft and equivalent piers Piled raft, refined mesh

Fig. 5. Mean settlement and load distribution of piled raft

7.5

320

CLANCY AND RANDOLPH

investigate the effects of pile±soil stiffness ratio Kps , aspect ratio Lp /dp and spacing ratio sp /dp . These studies show essentially identical differential settlement pro®les to those shown in Fig. 7. WORKED EXAMPLE (a)

(b)

Pile

Raft element

Assumed area of no raft~soil contact

Fig. 6. In¯uence of mesh re®nement on load distribution: (a) coarse mesh; (b) re®ned mesh

0

Settlement/mean settlement

Settlement/mean settlement

0.8

Normalized co-ordinate 0.5

1

To illustrate the application of the approximate methods described above, an example 15 3 15 piled raft system, using the properties given in Table 5, is now considered. The raft mesh used was similar to that shown in Fig. 4, but expanded for the greater number of piles (and including piles on the axes of symmetry), with ten rod ®nite elements per pile. The following sections give full calculations for each of the approximate methods in turn.

Normalized co-ordinate 0.5

0

0.9 1 1.1 1.2

Krs = 0.01

Krs = 0.01

Krs = 1.0

Krs = 10

0.8 0.9 1 1.1 1.2

Piled raft analysis Combined pile group and raft Raft and equivalent piers

Fig. 7. Displacement pro®les: Kps = 100; Lp /dp = 25; sp /dp = 5´0

1

321

DESIGN TOOLS FOR PILED RAFTS

Table 5. Properties for 15 3 15 piled raft worked example Soil

Pile

Es = 35 MPa ís = 0´4

Ep Lp dp sp

Raft

= 35 000 MPa = 20 m = 0´8 m =4m

Er ír tr Lr Br

Combined pile group and raft Randolph's (1983) approximate method represents the ®rst step in combining the pile group and raft, producing estimates of the mean settlement of the piled raft, and of the load sharing between the raft and pile group. Separate analyses of the raft and pile group in isolation were made to determine kp and kr : kp = 2´94 MN/mm; kr = 2´75 MN/mm. These values were substituted into equation (5) to produce an estimate of kpr , which was then used in equations (6) and (7) to calculate Pr /P and Pp / P. Table 6 gives the results of the analysis, and the corresponding values from the full analysis. Although the estimate for kpr is very close to that of the complete analysis, the proportion of load taken by the raft is overestimated by almost 20% of the total applied load. This follows the trend

= 35 000 MPa = 0´3 = 3´806 m = 60 m = 60 m

Dimensionless groups Lp /dp sp /dp Kps Lr /Br Krs

observed in the earlier study of a 10 3 10 piled raft system, using a similar level of mesh re®nement. It was impractical to produce results for the 15 3 15 piled raft using an increased level of mesh re®nement, due to the numerical size of the problem. Based on the results of the 10 3 10 analyses using different mesh re®nement levels, it is likely that the true load distribution lies between the combined and full analyses. To produce an estimate of the piled raft settlement pro®le, the method of factoring the raft settlements by kr /kpr (= 0´88 in this example) was used. The normalized settlements are plotted in Fig. 8, where the estimated pro®le is typical for Krs = 0´1. The results of the complete analysis, as with the 10 3 10 piled raft, show a slightly greater differential settlement.

Normalized co-ordinate

Settlement/mean settlement

0.8

0

= 25 =5 = 1000 =1 = 0´1

0.5

0.9

1

1.1

1.2

Piled raft analysis Combined plie group and raft Raft and equivalent piers

Fig. 8. Settlement pro®les for example 15 3 15 piled raft

1

322

CLANCY AND RANDOLPH

Table 6. Overall behaviour of 15 3 15 piled raft Combined pile group and raft

Equivalent piered raft

0´502

0´497

0´512

0´79 0´21

0´59 0´41

0´93 0´07

17

18

11

12

13

14

6

7

8

9

The performance of two piled raft foundations, 15 m apart, has been reported by Yamashita, Kakurai & Yamada (1994) and Yamashita & Kakurai (1991). The foundations were constructed in Urawa City, a suburb of Tokyo, and for identi®cation purposes will be referred to as foundation A and foundation B (see Fig. 9). In each case, a small number of piles (20 piles and 16 piles for foundations A and B respectively) at large spacings (typically sp /dp . 6) was used in an attempt to follow the recommendations of Cooke (1986) for settlement reducing piles. Each raft consists of foundation beams and foundation slabs; the beams are indicated by the shaded areas of Fig. 9.

20

15

10

12.0 m

23.0 m

1

2

3

6.0 m

Piles

4

6.0 m

6.0 m

5 6.0 m

24.0 m (a)

12

13

14

15

16

8

9

8 .4 m

10

4

5

6

6.3 m

7

2 6.3 m

5.3 m

7.9 m

4 .8 m

11

1

TWO CASE STUDIES

19

5.0 m

6.0 m

16

5.8 m

Raft and equivalent piers For the equivalent-piered raft, a 5 3 5 group of piers was used, where each pier represents a square 3 3 3 pile subgroup. The equivalent pier radius and modulus were calculated according to equations (9) and (10): rpeq = 4´51 m; Epeq = 2500 MPa. Comparative stiffness values for the 9-pile subgroup and the single pier were kp9 = 0´85 MN/mm and kpeq = 0´96 MN/mm. The equivalent pier overestimates the pile group stiffness by approximately 13%. Values of kpr , Pp /P and Pr /P for the complete system are given in Table 6 for comparison with the full piled raft analysis and the combined pile group and raft method. As with the 10 3 10 piled raft study, the equivalent pier approach overestimates the proportion of load taken by the pile group by at least 14% of the total load (and possibly by more if a more re®ned mesh is used for the full analysis). Figure 8 shows the predicted differential settlements from the equivalent piered raft approach, allowing direct comparison with the other two methods. As discussed above, analysis of the equivalent-piered raft allows the in¯uence of the pile group on differential settlements to be accounted for. Thus, the results are closer to the pro®le from a complete analysis than those from the combined pile group and raft approximation.

19.0 m

  P 1 knorm = . wpr np Es d p Pp /P Pr /P

Full piled raft analysis

3 6.3 m

6.3 m

4.2 m

29.4 m (b)

Fig. 9. Foundation plans for case study: (a) foundation A (Yamashita et al., 1994); (b) foundation B (Yamashita & Kakurai, 1991)

Soil properties Detailed soil pro®les for each foundation have been presented in the corresponding papers, and comprised alternating layers of variable-density sands, stiff clay and silt overlying dense sand at 50 m. The piles were founded in a layer of medium to dense sand in the depth range 16±19 m.

323

DESIGN TOOLS FOR PILED RAFTS

Although the foundations were adjacent, different soil properties were adopted in the publications of Yamashita et al. (1991) and Yamashita & Kakurai (1994). The originally cited soil properties have been assumed here, and are summarized in Tables 7 and 8. Raft properties For foundation A, no information was given for the thickness of the raft or the dimensions of the beams. For foundation B, the raft thickness was 300 mm and two sets of beams were identi®ed: all beams were of depth 1800 mm, the ®rst set were

relatively wide (350±900 mm) and the second set were relatively narrow (250±320 mm). Speci®c information on the dimensions of individual beams was not given; two representative widths, of 600 mm and 300 mm, were used. Foundation A covers a similar area to that of foundation B, and it was considered reasonable to assume a raft thickness of 300 mm and beam dimensions 600 3 1800 mm. The raft thicknesses quoted in Tables 7 and 8 are approximate equivalent values allowing for the beams, although these were actually modelled explicitly (see Fig. 10). Even with the beams, the raft±soil stiffness ratio is very low.

Table 7. Case study: foundation A property set Soil Es = 65´9 MPa ís = 0´3 zd = 48 m

Pile

Raft

Ep = 7850±11 740 MPa Lp = 16 m dp = 0´7±0´8 m sp < 5 m

Er = 20 600 MPa ír = 0´167 tr < 0´8 m Lr = 24 m Br = 23 m

Dimensionless groups Lp /dp < 22 sp /dp < 6´5 zd /dp < 65 Kps = 120±180 Lr /Br = 1´04 zd /Br = 2´1 Krs = 0´004

Table 8. Case study: foundation B property set Soil Es = 70´6 MPa ís = 0´4 zd = 63 m

Pile

Raft

Ep = 10 190±15 560 MPa Lp = 15 m dp = 0´5±0´6 m sp < 5 m

Er = 20 600 MPa ír = 0´167 tr < 0´65 m Lr = 29´4 m Br = 19 m

(a)

Fig. 10. Piled raft meshes for comparison with Yamashita et al. (1974)

Dimensionless groups Lp /dp < 27 sp /dp < 9 zd /dp < 115 Kps = 145±220 Lr /Br = 1´55 zd /Br = 3´3 Krs = 0´001

(b)

324

CLANCY AND RANDOLPH

Numerical analysis The small number of piles used in the foundations under consideration precludes the use of an equivalent piered raft. Thus the results of a complete piled raft analysis are compared with those of a combined pile group and raft. The raft ®nite element meshes used in the analyses, with increased raft thickness where the beams occur, are shown in Fig. 10. The predictions of overall foundation behaviour, in terms of settlement and load distribution, are given in Table 11, and compared with the measured values. For foundation A, the predictions of normalized stiffness are very close to the observed stiffness. The comparison of load distribution is not so favourable, with the greater discrepancy demonstrated by the combined pile group and raft approach. This is consistent with the earlier parametric study, where in all cases the combined approach overestimated the proportion of load carried by the pile group. The relatively large differences in this case may result from uncertainties in the raft stiffness, due to insuf®cient information on raft and foundation beam dimensions and any additional stiffening due to the

Pile properties The piles were constructed by inserting a steel H-section into a pre-augered borehole ®lled with soil cement. Seven different speci®cations were used; the pile and borehole dimensions for each are listed in Table 9 with reference to the pile numbers shown in Fig. 9. Only the breadth of the H-section was given for piles in foundation B, and the steel areas have been estimated from Japanese pile charts. In the present analysis, a circularsection pile of borehole dimensions was assumed, using an equivalent modulus derived from the steel alone (assuming the stiffness of soil cement to be negligible). Loading The design loading condition was for column loading at each pile location. In the present analysis, the loads were modi®ed to allow for a self-weight uniform load of 7 kPa due to the raft, while maintaining the same total applied load. The total design load of foundation A was 47´5 MN, and that of foundation B was 30´0 MN. The column loading distribution is given in Table 10.

Table 9. Case study: pile speci®cations Borehole diameter: mm

H-pile dimensions: mm

Equivalent Ep : MPa

Piles

Foundation A 800

414 3 405 3 18 3 28

11 740

A7, A8, A9

800

400 3 400 3 13 3 21

8710

A2, A3, A4, A6, A10, A12, A14

700

350 3 350 3 12 3 19

9040

A1, A5, A11, A15

700

300 3 300 3 10 3 15

7850

A13, A16, A17, A18, A19, A20

Foundation B 600

400

15 560

B5, B6, B7, B12, B13, B14

500

300

12 220

B1, B2, B3, B4, B10, B11

500

250

10 190

B8, B9, B15, B16

Table 10. Cast study: applied column loads Pile

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Foundation A Load: MN

1´98 2´83 2´71 2´49 1´72 2´56 3´54 3´62 3´45 2´60 2´05 2´36 1´45 2´67 1´48 1´15 1´64 0´94 1´44 0´88 Foundation B

Load: MN

1´40 1´17 0´95 1´83 3´49 3´94 2´70 1´27 0´38 1´07 0´66 2´25 2´22 2´00 0´97 0´51

325

DESIGN TOOLS FOR PILED RAFTS

Table 11. Case study: overall foundation behaviour Full piled raft analysis

Combined pile group and raft

Measured

Foundation A knorm (de®ned in Table 7)

3´17

3´23

,3´2

Pp /P

0´66

0´74

0´49

Pr /P

0´34

0´26

0´51

knorm

4´54

4´43

Pp /P

0´49

0´60

0´51

Pr /P

0´51

0´20

0´49

Foundation B

superstructure. In addition, the measured load distribution underestimated the proportion of load carried by the piles because the instrumentation was capable only of recording the load in the steel H-section. Yamashita et al. (1994) estimated this error to be within about 8%. Fewer uncertainties existed in the analysis of foundation B, where more information was available for the raft and foundation beams. In this case, the numerical methods underpredicted the normalized foundation stiffness by about 25%; however, the `measured' value is an estimate, judged from the maximum and minimum recorded settlements. The load distribution ratios calculated from a complete piled raft analysis were very close to the measured values, but again the combined pile group and raft approach overestimated the proportion of load in the pile group. As with foundation A, the measured pile load was only that portion carried by the steel H-section, and is therefore an underestimate. The true load distribution is expected to lie between the values of the two numerical methods. Settlement contours are shown in Fig. 11, where the results of a factored raft analysis are also plotted. The contour plots for foundation A do not compare very well, which is most likely due to the distribution of piles under the raft. However, the plots highlight an interesting point: by providing excessive support at the edge of the raft, the piles have actually led to a relative increase in differential settlement. This is the reverse of the pile distribution recommended by Randolph & Clancy (1993), where it was suggested that a small number of piles should be concentrated in the central 25% of the raft area. For foundation B, the comparison is much closer and little difference is evident between the two methods. Table 12 gives comparisons between predicted

,6´0

and measured values of settlement and pile load. The combined pile group and raft approach allows only comparison of settlements, while the full piled raft analysis can provide details of load distribution. Both foundation A and foundation B demonstrate greater differential settlements than predicted; in each case the full piled raft analysis shows a closer comparison. The predicted pile loads are signi®cantly higher than those measured for foundation A. Correspondingly, the average computed raft pressures are a factor of two lower than measured for foundation A, but within 5% of the measured values for foundation B. Overall, considering the uncertainties involved in the present analysis and in the ®eld measurements, the results are reasonable. CONCLUSIONS

Two alternative approximate methods for the analysis of piled raft foundations have been presented. The ®rst method combines the separate responses of the pile group and raft in isolation; the second uses an equivalent-piered raft approach to reduce greatly the number of piles analysed. Both methods allow the calculation of overall piled raft stiffness, load-sharing between the pile group and raft, and estimates of differential settlement. Comparisons have been made between the approximate methods and a complete analysis for piled rafts containing up to 225 piles. Estimates of overall foundation stiffness were in excellent agreement for all methods. Differences arose in the relative proportion of load carried by the piles and by the raft, partly due to insuf®ciently ®ne discretization of the raft in the full analysis, which led to overprediction of the proportion of load carried by the pile group. Load distribution predictions from the equivalent-piered

326

CLANCY AND RANDOLPH 10

12

11

10

12

9 15

18

6 13 10 9

12

10

12 (c) 15 (a)

13 13

10

10

15 16

6 13

15 13

10

13

9

15 (d) 16 (b)

Fig. 11. Settlement contours (mm): (a, b) foundation A; (c, d) foundation B; (a, c) piled raft analyses; (b, d) raft analyses factored by kr /kpr

Table 12. Case study: pile loads (MN) and settlement (mm) Foundation A Pile Measured load Predicted load Pile Measured settlement Predicted settlement Predicted settlement{

1

2

3

6

7

8

11

Foundation B 12

13

16

18

20

2

6

13

0´62 1´47 1´35 1´32 1´70 0´51 1´26 1´26 1´16 0´15 0´71 0´64 0´53 2´08 1´35 1´81 1´98 2´03 1´93 1´93 1´99 1´74 1´43 1´05 1´14 1´11 1´01 0´87 1´39 1´46 1

3

5

6

8

10

12´5 12´5 15´0 12´5 16´3 17´5

16

18

20

4

7

11

14

7´5

7´5

7´5

3´0

10´5

8´0

3´0

11´9 15´0 11´1 15´1 18´3 14´6 11´3 13´5 10´2

10´5 12´5

16´5 16´8 15´0 16´1 16´9 15´4 14´5 13´7 12´7

10´7 13´5 10´6 11´8

 Full piled raft analysis { Combined pile group and raft

8´3 10´4

DESIGN TOOLS FOR PILED RAFTS

raft approach overestimated the proportion of load taken by the pile group, partly through overprediction of the pile stiffness using an equivalent pier, and partly due to the greater raft area occupied by the equivalent pier. Normalized settlement pro®les were generally in very good agreement, with the greatest differences arising for the combined approach with raft stiffness Krs = 0´1, where differential settlements were overestimated by up to 5% by the approximate methods. Two case studies were described, based on piled raft foundations in the Urawa City suburb of Tokyo. The pile group of each foundation contained only a small number of piles, facilitating a complete piled raft analysis, but precluding the use of an equivalent pier approach. Numerical analysis of the ®rst foundation (foundation A) indicated some limitations in the combined pile group and raft approach for predicting differential settlements; the method was unable to model the uneven distribution of support provided by the widely spaced piles. The predictions of overall foundation behaviour were reasonable, considering the uncertainties in foundation geometry. The second foundation (foundation B) had been documented in greater detail, allowing a closer numerical representation. In this case, both of the numerical analyses provided good predictions of overall foundation behaviour. Again there was a tendency to underestimate differential settlement, but closer predictions were made of pile load distribution. This paper demonstrates that approximations for piled raft analysis can produce useful results for design purposes. The approximate methods are simple to apply and require low time and computational resources. However, where possible, ®nal designs should be checked by a full piled raft analysis. ACKNOWLEDGEMENTS

The ®rst author gratefully acknowledges ®nancial support during the work reported here, through an Overseas Postgraduate Research Scholarship from the Australian Department of Employment, Education and Training, and a University Research Scholarship from the University of Western Australia.

NOTATION Ag plan area of a pile group Ap sum of pile cross-sectional areas dp pile diameter Epeq equivalent pier modulus kp overall stiffness of a piled group kpr overall stiffness of a piled raft kr overall stiffness of a raft in isolation

327

pile±soil stiffness ratio embedded length of a pile ratio of circular raft diameter to pile diameter total number of piles in a group load carried by a pile group load carried by a raft radius of in¯uence of a pile equivalent pier radius factor used in plotting normalized differential settlement árp in¯uence of a pile group on the raft ís Poisson's ratio of soil r soil inhomogeneity factor

Kps Lp n np Pp Pr rm rpeq R

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of American Society of Civil Engineers Geotechnical Engineering Division specialty conference, Texas. New York: American Society of Civil Engineers. Smith, I. M. & Grif®ths, D. V. (1988). Programming the ®nite element method, 2nd edn. Chichester: Wiley. Yamashita, K. & Kakurai, M. (1991). Settlement behaviour of the raft foundation with friction piles. Proc. 4th Int. Conf. Piling Deep Fdns, Stressa, 461±466. Yamashita, K., Kakurai, M. & Yamada, T. (1994). Investigation of a piled raft foundation on stiff clay. Proc. 13th Int. Conf. Soil Mech., New Delhi 2, 543± 546.